Knowledge of material included in the undergraduate Analysis course (110.405-406) is assumed.  This includes topics such as open and closed sets, compactness (incl. Heine-Borel theorem), continuity (incl. uniform continuity and the Arzela-Ascoli theorem). 

     1. Measure theory:  Measurable sets and functions, outer measure, construction of Lebesgue measure. Lusin's theorem. Notions of convergence involving measure (pointwise a.e., convergence in measure). Egorov's theorem.

       2. Lebesgue integral: Definition of $\int_X f(x) d \mu(x)$  and relation to the distribution function $\mu\{x: f(x) > t\}$.  Fatou's Lemma and dominated convergence. $L^1(X, d\mu)$ as a normed space. Relation of convergence in $L^1$ to pointwise a.e. convergence and convergence in measure. Product measure and the Fubini theorem.  Relation  between Lebesgue  and Riemann integrals.  Riesz representation theorem concerning positive linear functionals on $C(X)$. 

      3. Lp spaces: Jensen, Holder and Minkowski inequalities. Completeness. Duality of $L^p$ and $L^q$ for $\frac{1}{p} + \frac{1}{q} = 1$. Bounded functionals, weak convergence on $L^p$. Uniform boundedness principle for $L^p(X, \mu)$. Approximation of $L^p$ functions on $\R^n$ by smooth functions. Weak compactness of unit ball in $L^p$. $L^2(X, d\mu)$.  Hilbert space: Bessel inequality, orthonormal bases. 

      4. Fourier analysis: Fourier transform ${\mathcal F}$ on $L^1(\R^n)$ and $L^1(S^1)$ ($S^1$ = unit circle). Riemann-Lebesgue lemma. Plancherel theorem (${\mathcal F}$ as a unitary operator on $L^2(\R^n)$). Parseval  inequality for ${\mathcal F}$ on $S^1$. Inversion formula. Convolution. Hausdorff-Young inequality.

W. Rudin, Real and Complex Analysis, McGraw Hill. 

E. Lieb and M. Loss, Analysis, AMS Graduate Studies in Math. 14.

 G. B. Folland, Real Analysis, Wiley Interscience.

1. Basic metric space topology: Compact, connected, boundary, simply connected, uniform convergence, Arzela-Ascoli theorem.
2. Analytic (= holomorphic) functions: Power series, radius of convergence, Cauchy-Riemann equations. Liouville theorem, Cauchy estimates. Uniform limits. Discreteness of zeros.
3. Meromorphic functions: Riemann’s removable singularities theorem, poles, Laurent series, Cauchy integral formula, residue theorem, residue calculus.
4. Local behaviour of holomorphic functions: Argument principle, zeros of holomorphic functions, Rouche’s theorem, Hurwitz’s theorem. Maximum modulus principle. Open mapping theorem.
5. Holomorphic mappings: Riemann sphere, linear fractional transformations, conformal mapping, Schwarz lemma, normal families, Riemann mapping theorem.

Any one of the following texts should amply cover the material on the exam.

Introduction to Complex Analysis by J. Noguchi

Function Theory of One Complex Variable by R.E. Greene and S.G. Krantz

Functions of One Complex Variable by J.B. Conway

Complex Analysis by L. Ahlfors